A widely referenced source on a rigorous coupled wave (RCW) algorithm is that of Moharam and Gaylord (M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, J. Opt. Soc. Am. A, Vol. 12, No. 5, p. 1068 (1995)). A schematic from their paper showing the diffraction problem is reproduced in FIG. 1. In particular, FIG. 1 defines the various incident conditions for the diffraction problem. The plane of incidence is defined by the polar angle, theta, and the azimuthal angle, phi. The azimuthal angle defines the angle the incident plane makes with the plane perpendicular to the grating lines, so that phi=0 corresponds to the classical incidence case. The angle psi defines the direction of the electric field with respect to the plane of incidence, with psi=90 corresponding to s polarized and psi=0 to p polarized incident light.
As shown in FIG. 1, a diffraction grating 100 has a grating region 104 formed in a substrate 102 (the substrate being designated as Region II). The grating region 104 has a height d as indicated in the figure. Region I is comprised of the material above the grating region, in this case in air or vacuum space. As indicated in FIG. 1, the grating region may be formed of alternating grating lines 106 and grating spaces 108. The grating lines 106 may have a width 109. The grating periodicity is characterized by the grating period 110 as indicated.
It will be recognized that a diffraction grating may be formed in other manners than that of FIG. 1 and that FIG. 1 is only one exemplary diffraction grating as known to those skilled in the art. For example, a diffraction grating need not be formed utilizing spaces. FIG. 1B shows one such alternative diffraction grating. As shown in FIG. 1A, the diffraction grating 100 may be comprised of grating lines 106A and 106B, again having a grating region 104 with height d. In this example, grating lines 106A and 106B will be formed in a manner in which the lines have different optical properties. Further, though the examples shown include gratings having two different optical properties within each period of the diffraction grating, it will be recognized that the diffraction grating may comprise three or more different materials within each period. Likewise, though each grating line is shown as a single material, it will be recognized that the grating lines may be formed of multiple layers of the same or different materials. In addition, though the grating lines are shown as being “squared off,” it will be recognized that each line may have sloped sides, curved edges, etc.
With reference again to FIG. 1, an x-y-z coordinate system is shown having a frame of reference in which the x-direction is shown as being perpendicular to the original alignment of the grating lines. The plane of incidence 112 of the incident light is defined by the polar angle 114, theta and the azimuthal angle 116, phi. The electric field 120 has a propagation vector 122 (k) of the incident wave. The unit vectors 124 (t) and 126 (n) are tangent and normal to the plane of incidence, respectively. As mentioned above, the angle 128, psi, defines the direction of the electric field with respect to the plane of incidence.
The RCW method involves the expansion of the field components inside and outside the grating region in terms of generalized Fourier series. The method consists of two major parts—an eigen-problem to determine a general solution inside the grating layer, and a boundary problem to determine the reflected and transmitted diffracted amplitudes along with the specific solution for the fields inside the grating region. The Fourier series are truncated after a finite number of terms. The truncation is usually characterized by the truncation order, N, which means that 2N+1 spatial harmonics are retained in the series (positive and negative terms to ±N, and the 0 term).
Standard methods for solving the eigen-problem, boundary problem, and the various other matrix multiplications and inversions involved are order N3 operations. This means that an increase of the truncation order by a factor of two results in an increase in overall computation time by a factor of approximately 8. The truncation order required for convergence is determined by the specifics of the diffraction problem, and generally increases for larger pitch to incident wavelength ratios and larger optical contrast between grating lines and spaces. The result is that while some diffraction problems are very tenable, others quickly become impractical to solve due to a large computation cost.
In the case of the phi=0 classical mount, the diffraction problem decouples into TE and TM components, which can be solved separately (for the phi=0 mount, TE polarization corresponds to s polarized incident light, and TM polarization corresponds to p polarized incident light). Any arbitrary polarization is decomposed into a combination of the TE and TM problems. In practice, the incident light is often purely TE or TM polarized, and only one case needs to be solved. For given truncation order N and classical mount the eigen-problem is of size 2N+1, and the boundary problem is of size 2(2N+1).
The general case where phi≠0 is known as conical diffraction. In this case, the s and p components are coupled, with a corresponding increase in the amount of computation time. The boundary problem involves 4(2N+1) sized matrices. The eigen-problem has been successfully decoupled into two smaller eigen-problems, each of size 2N+1 (see Moharam and Gaylord 1995 referenced above, or S. Peng and G. M. Morris, J. Opt. Soc. Am. A, Vol. 12, No. 5, p. 1087 (1995)). Therefore, the computation time for the general conical incidence case suffers a factor of 2 increase for the eigen-problem and a factor of 8 increase for the boundary problem compared to the corresponding classical mount case with same polar incident angle, theta.
Analysis of a diffraction grating problem is of particular use to determining the various characteristics of the diffraction grating structure. For example, critical dimensions of a device (such as in semiconductor processing in one exemplary use) may be monitored by evaluating the characteristics of a diffraction grating as is known in the art. By evaluating data from known optical metrology tools using regression and/or library methods, the diffraction analysis may lead to, for example, a determination of the grating line widths, the grating height/depth, the period of the grating, the slopes and profiles of the grating, the material composition of the grating, etc. As known in the art, such grating characteristics may be related to the characteristics of a device that is being analyzed, such as for example but not limited to widths, heights, depths, profiles, etc. of transistors, metallization lines, trenches, dielectric layers, or the like, all as is known to those skilled in the art. Since the regression and/or library methods may require many calculations of diffraction efficiencies, special consideration must be given to computation expense in such applications.